Sufficient conditions for the convexity of the level sets of ground-state solutions

Abstract.

Let Ω be a bounded convex domain in $\mathbb{R}^{n}$. We consider constrained minimization problems related to the Euler-Lagrange equation

$$ -\Delta u + V (x)u = \lambda u^{p}\,{\rm in } \Omega , u > 0 , $$

over classes of functions ${u \in H^{1}_{0}}$ (Ω) with convex super level sets. We then search for sufficient conditions ensuring that the minimizer obtained is a classical solution to the above equation.